- fourier basis in orthogonal, but not orthonormal
- many natural and man-made phenomena exhibit an oscillatory behavior
- after a certain amount of time, called the period, these phenomena comes back to the same position
- conversely, we can define the frequency, i.e. the inverse of the period, that indicates the number of oscillations of the system per second
- it thus make sense to use sines and cosines as basic building blocks to represent these oscillatory signals
- this is the basic goal of Fourier analysis: to decompose a signal in terms of sines and cosines
- two kinds of fourier tools,
- fourier analysis:
- allows determining the frequency components present in a signal
- We move from the time to the frequency domain
- fourier synthesis:
- allows constructing signals with known frequency components
- we move from the frequency domain to the time domain
- We have started our exploration of fourier analysis with the simplest tool
- the discrete Fourier transform (DFT) that applies to finite-length signals
- if N is the length of the signals, we have seen that the set of complex exponential
- \(w_k[n]=e^{j\frac{2\pi}{N}nk}\)
- \( n=0,…,˙N−1,k=0,…,N−1 \)
- with fundamental frequency indexed by k,
- \( \frac{2\pi}{N}k N 2πk, \)
- forms an orthogonal basis of \( \mathbb{ℂ}^N \). The DFT is just a change of basis in this space.